PhD Thesis

Perfect sampling algorithms are Markov Chain Monte Carlo
(MCMC) methods without statistical error. The latter are used when one
needs to get samples from certain (non-standard) distributions. This can be
accomplished by creating a Markov chain that has the desired distribution
as its stationary distribution, and by running sample paths "for a long
time", i.e. until the chain is believed to be in equilibrium. The
question "how long is long enough?" is generally hard to answer and the
assessment of convergence is a major concern when applying MCMC schemes.
This issue completely vanishes with the use of perfect sampling
algorithms which - if applicable - enable exact simulation from the
stationary distribution of a Markov chain.

In this thesis, we give an introduction to the general idea of MCMC
methods and perfect sampling. We develop advances in this area and
highlight applications of these advances to two relevant problems.

As advances, we discuss and devise several variants of the well-known
Metropolis-Hastings algorithm which address accuracy, applicability, and
computational cost of this method. We also describe and extend the idea of
slice coupling, a technique which enables one to couple continuous sample paths
of Markov chains for use in perfect sampling algorithms.

As a first application, we consider Bayesian variable selection.
The problem of variable selection arises when one wants to model the
relationship between a variable of interest and a subset of explanatory
variables.
In the Bayesian approach one needs to sample from the posterior distribution
of the model and this simulation is usually carried out using regular MCMC
methods.
We significantly expand the use of perfect sampling algorithms within this
problem using ideas developed in this thesis.

As a second application, we depict the use of these methods for
the interacting fermion problem. We employ perfect sampling for computing self
energy through Feynman diagrams using Monte Carlo integration techniques.